Static and Dynamic Buckling of Rectangular Functionally Graded Plates Subjected to Thermal Loading
In the paper, the buckling phenomenon for static and dynamic loading (pulse of finite duration) of functionally graded plates subjected to uniform temperature increment is presented. The work deals with thin rectangular plates with unmovable edges, simply supported or clamped along all edges....
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
2013
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irk-123456789-1126762020-12-17T17:39:36Z Static and Dynamic Buckling of Rectangular Functionally Graded Plates Subjected to Thermal Loading Czechowski, L. Kowal-Michalska, K. Научно-технический раздел In the paper, the buckling phenomenon for static and dynamic loading (pulse of finite duration) of functionally graded plates subjected to uniform temperature increment is presented. The work deals with thin rectangular plates with unmovable edges, simply supported or clamped along all edges. The material properties varying smoothly across the thickness are assumed to be temperature-independent. The investigations are conducted for different values of volume fraction index and uniform temperature rise in form of rectangular pulse of finite duration. Представлено явление потери устойчивости при статической и динамической нагрузке (импульс конечной длительности) функционально-градиентных пластин, подверженных воздействию равномерного повышения температуры. Рассмотрены шарнирно опертые или закрепленные (вдоль кромок) тонкие прямоугольные пластины с неподвижными кромками. Предполагается, что свойства материала, плавно изменяющиеся по толщине, не зависят от температуры. Проведены исследования для разных значений коэффициента относительного объема и равномерного приращения температуры, представленного в виде прямоугольного импульса конечной длительности. Представлено явище втрати стійкості під час статичного і динамічного навантаження (імпульс скінченної тривалості) функціонально-градієнтних пластин за рівномірного підвищення температури. Розглянуто шарнірно оперті або закріплені (вздовж кромок) тонкі прямокутні пластини з нерухомими кромками. Припускається, що властивості матеріалу, які плавно змінюються по товщині, не залежать від температури. Проведено дослідження для різних значень коефіцієнта відносного об’єму і рівномірного приросту температури, представленого у вигляді прямокутного імпульсу скінченної тривалості. 2013 Article Static and Dynamic Buckling of Rectangular Functionally Graded Plates Subjected to Thermal Loading / L. Czechowski, K. Kowal-Michalska // Проблемы прочности. — 2013. — № 6. — С. 45-55. — Бібліогр.: 16 назв. — англ. 0556-171X http://dspace.nbuv.gov.ua/handle/123456789/112676 539.4 en Проблемы прочности Інститут проблем міцності ім. Г.С. Писаренко НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
| topic |
Научно-технический раздел Научно-технический раздел |
| spellingShingle |
Научно-технический раздел Научно-технический раздел Czechowski, L. Kowal-Michalska, K. Static and Dynamic Buckling of Rectangular Functionally Graded Plates Subjected to Thermal Loading Проблемы прочности |
| description |
In the paper, the buckling phenomenon for static
and dynamic loading (pulse of finite duration)
of functionally graded plates subjected to uniform
temperature increment is presented. The
work deals with thin rectangular plates with unmovable
edges, simply supported or clamped
along all edges. The material properties varying
smoothly across the thickness are assumed to be
temperature-independent. The investigations are
conducted for different values of volume fraction
index and uniform temperature rise in form
of rectangular pulse of finite duration. |
| format |
Article |
| author |
Czechowski, L. Kowal-Michalska, K. |
| author_facet |
Czechowski, L. Kowal-Michalska, K. |
| author_sort |
Czechowski, L. |
| title |
Static and Dynamic Buckling of Rectangular Functionally Graded Plates Subjected to Thermal Loading |
| title_short |
Static and Dynamic Buckling of Rectangular Functionally Graded Plates Subjected to Thermal Loading |
| title_full |
Static and Dynamic Buckling of Rectangular Functionally Graded Plates Subjected to Thermal Loading |
| title_fullStr |
Static and Dynamic Buckling of Rectangular Functionally Graded Plates Subjected to Thermal Loading |
| title_full_unstemmed |
Static and Dynamic Buckling of Rectangular Functionally Graded Plates Subjected to Thermal Loading |
| title_sort |
static and dynamic buckling of rectangular functionally graded plates subjected to thermal loading |
| publisher |
Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| publishDate |
2013 |
| topic_facet |
Научно-технический раздел |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/112676 |
| citation_txt |
Static and Dynamic Buckling of Rectangular Functionally Graded Plates Subjected to Thermal Loading / L. Czechowski, K. Kowal-Michalska // Проблемы прочности. — 2013. — № 6. — С. 45-55. — Бібліогр.: 16 назв. — англ. |
| series |
Проблемы прочности |
| work_keys_str_mv |
AT czechowskil staticanddynamicbucklingofrectangularfunctionallygradedplatessubjectedtothermalloading AT kowalmichalskak staticanddynamicbucklingofrectangularfunctionallygradedplatessubjectedtothermalloading |
| first_indexed |
2025-07-08T04:23:32Z |
| last_indexed |
2025-07-08T04:23:32Z |
| _version_ |
1837051269489885184 |
| fulltext |
UDC 539.4
Static and Dynamic Buckling of Rectangular Functionally Graded
Plates Subjected to Thermal Loading
L. Czechowski
1
and K. Kowal-Michalska
2
Lodz University of Technology, Lodz, Poland
1 leszek.czechowski@p.lodz.pl
2 katarzyna.kowal-michalska@p.lodz.pl
ÓÄÊ 539.4
Ñòàòè÷åñêàÿ è äèíàìè÷åñêàÿ ïîòåðÿ óñòîé÷èâîñòè ïðÿìîóãîëüíûõ
ôóíêöèîíàëüíî-ãðàäèåíòíûõ ïëàñòèí, ïîäâåðæåííûõ òåïëîâîé
íàãðóçêå
Ë. ×åõîâñêè, Ê. Êîâàëü-Ìèõàëüñêà
Ëîäçèíñêèé òåõíè÷åñêèé óíèâåðñèòåò, Ëîäçü, Ïîëüøà
Ïðåäñòàâëåíî ÿâëåíèå ïîòåðè óñòîé÷èâîñòè ïðè ñòàòè÷åñêîé è äèíàìè÷åñêîé íàãðóçêå
(èìïóëüñ êîíå÷íîé äëèòåëüíîñòè) ôóíêöèîíàëüíî-ãðàäèåíòíûõ ïëàñòèí, ïîäâåðæåííûõ âîç-
äåéñòâèþ ðàâíîìåðíîãî ïîâûøåíèÿ òåìïåðàòóðû. Ðàññìîòðåíû øàðíèðíî îïåðòûå èëè
çàêðåïëåííûå (âäîëü êðîìîê) òîíêèå ïðÿìîóãîëüíûå ïëàñòèíû ñ íåïîäâèæíûìè êðîìêàìè.
Ïðåäïîëàãàåòñÿ, ÷òî ñâîéñòâà ìàòåðèàëà, ïëàâíî èçìåíÿþùèåñÿ ïî òîëùèíå, íå çàâèñÿò îò
òåìïåðàòóðû. Ïðîâåäåíû èññëåäîâàíèÿ äëÿ ðàçíûõ çíà÷åíèé êîýôôèöèåíòà îòíîñèòåëüíîãî
îáúåìà è ðàâíîìåðíîãî ïðèðàùåíèÿ òåìïåðàòóðû, ïðåäñòàâëåííîãî â âèäå ïðÿìîóãîëüíîãî
èìïóëüñà êîíå÷íîé äëèòåëüíîñòè.
Êëþ÷åâûå ñëîâà: äèíàìè÷åñêîå âûïó÷èâàíèå, ôóíêöèîíàëüíî-ãðàäèåíòíûå
ïëàñòèíû, òåïëîâàÿ íàãðóçêà.
Introduction. Known since 1984 functionally graded materials (FGM) very
soon became popular in scientific research and engineering applications. Typical
FGM is an inhomogeneous composite made up of two constituents – typically of
metallic and ceramic phases which relative content changes gradually across the
thickness of a plate or a shell. This eliminates the adverse effects between the
layers (e.g., shear stress concentrations and/or thermal stress concentrations),
typical for layered composites. The high-resistance heat capacity of ceramic and
good mechanical properties of metal phase make that the leading application area
of FGM structures are high-temperature environments (spacecraft, nuclear reactors
or structures for the chemical industry and defence) [1, 2].
Nonlinear analysis of plates and shells devoted to basic types of loads is
covered in Shen monograph [2]. He considered static bending and thermal bending
as an introduction to buckling and postbuckling behaviour of FGM plates and
shells. The shear deformation effect is employed in the framework of Reddy’s
higher order shear deformation theory (HSDT).
© L. CZECHOWSKI, K. KOWAL-MICHALSKA, 2013
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 6 45
In [1], alongside HSDT for FGM plates Reddy presents the comparison of
first order shear deformation theory (FSDT) and classical laminate plate (CLPT)
theory application for functionally graded plates. According to presented results it
is obvious that for thin plates the application of FSDT gives results in practice the
same as HSDT. The discrepancy between both theories is of 2% in calculated
deflections of analyzed plates.
The static buckling problem of functionally graded plates is discussed in the
frame of different approaches, e.g., in [3–5] – biaxial in-plane compression and
thermal loads (constant temperature) with axial compression, in [6, 7] – biaxial
in-plane compression, in [8] – for thermal stresses only, and in [9] – for different
temperature distribution through the plate thickness.
In mentioned above publications the dominant subject are the static mechanical
or steady-state thermal loadings. The dynamic types of analyses concern mostly
the vibrations problems, e.g., [10]. From authors previous experience connected
with static and dynamic analysis of thin-walled isotropic and orthotropic composite
plates and columns, e.g., [11, 12] the dynamic buckling of thin-walled structure is a
theoretically difficult problem but of great importance for practical engineering
applications.
It is well known that for pulse loads of short duration the structure can
withstand the dynamic loading magnitude much greater than the static one. The
dynamic pulse buckling occurs for pulses of intermediate amplitude and duration
close to the period of fundamental natural flexural vibration. In such case the
effects of damping are neglected. It should be remembered that opposite to the
static loading, the dynamic buckling only occurs for imperfect structure, the
bifurcational dynamic buckling load does not exist. Therefore, it is necessary to
define this ‘critical’ load on the basis of an assumed dynamic buckling criterion. In
most publications the Budiansky–Hutchinson criterion [13] is applied to determine
the dynamic critical load – it means the amplitude of pulse load which at given
duration causes the dynamic buckling.
Dynamic buckling criterion of Budiansky–Hutchinson states that: dynamic
stability loss occurs, when the maximal plate deflection grows rapidly with the
small variation of the load amplitude.
Therefore, similarly to the mechanical loading, the parameter DLF is
introduced as a quotient of pulse amplitude of temperature increment to the static
critical temperature rise:
DLF T Tdyn
cr
st� � � . (1)
The critical value of dynamic load factor DLFcr is determined on the basis of
the assumed criterion and then the critical value of dynamically applied temperature
increment �Tcr
dyn can be found.
In previous works [14, [15] static and dynamic buckling analysis was
presented for FG rectangular simply supported plates under mechanical (uniform
compression) and thermomechanical loading. In this paper the dynamic response
of thin FG rectangular plates, subjected to an uniform temperature rise in form of
an rectangular pulse of finite duration is analysed basing on the finite element
method (ANSYS software).
L. Czechowski and K. Kowal-Michalska
46 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 6
1. Modeling of Material Properties. The studied plates have continuously
varying material properties only in the thickness direction. On the top surface of a
plate, pure metal is assumed, grading finally to the bottom surface containing pure
ceramics. The volume fractions of metal Vm and ceramics Vc can be written
accordingly to the simple power law distribution:
V z
z
hm
n
( ) ,� �
�
�
�
�
1 (2)
V z V zc m( ) ( ),� �1 (3)
where 0� �z h, n denotes the volume fraction index across the plate thickness h
( , ),n �
�0 for n�0 plate is metallic and for n�� is ceramic.
Figure 1 presents the distribution of volume fraction of ceramic and metal
components across the plate thickness for different values of index n.
According to the law of mixtures, the effective material properties Peff (it
means: � is density, � is coefficient of thermal expansion, E is the Young
modulus, and � is Poisson’s ratio), are obtained in a form
P z P V z P V z P P P
z
heff m m c c c m c
n
( ) ( ) ( ) ( ) ,� � � � � �
�
�
�
�
1 (4)
where Pc i Pm denote material properties of ceramics and metal, respectively,
assumed as temperature independent.
2. Subject of Consideration. Thin, rectangular plates made of functionally
graded material were considered. Two different boundary conditions were taken
into account: all edges clamped and all edges simply supported. In both cases the
edges are immovable (see Table 1).
Static and Dynamic Buckling of Rectangular Functionally Graded Plates ...
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 6 47
Fig. 1. Volume fraction of ceramics and metal dependent on the distribution index across plate
thickness.
An uniform temperature rise through the whole plate and continuously
varying material properties across the plate thickness related to the power law
distribution were assumed in the investigations. When dynamic buckling is
investigated, the temperature increment is applied in form of rectangular pulse of
value �T dyn and duration time t p .
3. Numerical Model. The numerical simulations and appropriate calculations
have been conducted using the finite element method (ANSYS14.0).
The numerical models assigned for study were divided into SHELL132 and
SHELL281 type elements [16] (Fig. 2). SHELL132 has eight nodes with up to 32
temperature degrees of freedom at each node and is applicable to a 3D steady-state
or transient thermal analysis. SHELL281 has also eight nodes with six degrees of
freedom at each node and includes the effects of transverse shear deformation. This
applied element is well-suited for linear, high-rotation or high-strain nonlinear
applications and moderately thick shell structures. In addition, SHELL281 element
is very useful for multilayered applications to model laminated composite shells or
sandwich structures what in case of functionally graded materials across the
thickness is very profitable.
A shell element containing 20-layers with various material properties was
considered. For a determination of large deflections of structures the Green–
Lagrange nonlinear-displacement equation was employed. Calculations were
incrementally performed using the Newton–Raphson convergence approximation
method. The time step in applied Newmark time integration procedure has been
taken as 1/50 of the period of plate fundamental natural vibration.
Numerical simulation was carried out in two stages. In the first stage it was
essential to determine static critical temperature rise and corresponding buckling
mode and also a frequency of fundamental flexural natural vibrations for each
structure according to assumed distribution of material properties and boundary
conditions. Next, calculations in incremental solution for large strains taking into
account the pulse duration corresponding to period of natural flexural vibrations of
a given structure were performed. For all considered structures the initial deflection
corresponding to the first buckling mode of amplitude has w h0 0 01� . been taken
into account. The observation of structures behavior was continued in time two
times longer than the pulse duration.
48 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 6
L. Czechowski and K. Kowal-Michalska
T a b l e 1
Boundary Conditions of Plates
Clamped plate Simply supported plate
Edge 1, 2, 3, 4:
u u ux y z x y z, , , , ,rot rot rot � 0
Edge 1, 3: u ux z, � 0
Edge 2, 4: u uy z, � 0
4. Results of Numerical Calculations. The calculations were performed for
FG square plates of ratio width to thickness a h�100. The material properties of
components were assumed as shown in Table 2. The duration t p of the temperature
pulse was equal to one or one half of period of natural flexural vibrations t0 of
considered structure with regard to boundary conditions and material properties
distribution across the plate thickness (see Table 3).
The results of calculations have been presented in diagrams as the relation
between non dimensional maximal deflection (w h) and the applied temperature
increment (statically �T st or as a pulse loading �T dyn ) and additionally as the
relation w h versus DLF .
In aim to compare the correctness of the applied model the results obtained in
static case have been compared with the results known from the literature [9]. The
very good agreement has been achieved for the critical values of temperature rise
DTcr
st and also for postbuckling behavior (exemplary curves are shown in Fig. 3).
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 6 49
Static and Dynamic Buckling of Rectangular Functionally Graded Plates ...
a
b
Fig. 2. Types of applied elements with 8 nodes: Shell132 (a) and Shell281 (b).
The results of numerical calculations for chosen values of volume fraction
index n and two values of pulse duration of temperature increment have been
shown in Figs. 4 and 5 for plates with clamped edges and in Figs. 6 and 7 for plates
simply supported along all edges. Static and dynamic critical values of temperature
increment have been collected in Table 4 (for pulse duration equal to the period of
natural flexural vibrations t tp � 0) and in Table 5 (for pulse duration equal to the
period of natural flexural vibrations t tp �0 5 0. ).
50 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 6
L. Czechowski and K. Kowal-Michalska
T a b l e 2
Material Properties of Constituents
Properties Al2O3 Ni
Young’s modulus E, GPa 393 200
Thermal expansion coefficient �, K�1 13 3 10 6. � � 8 8 10 6. � �
Poisson’s ratio � 0.25 0.3
Density �, kg/m3 2000 7800
T a b l e 3
Periods of Natural Flexural Vibrations t0 (ms)
Volume fraction index n Clamped plate Simply supported plate
0 11.20 20.80
0.3 9.19 17.07
1.0 7.62 13.80
5.0 5.50 11.04
� 4.18 7.65
Fig. 3. Static post buckling curves for metal plates subjected to uniform temperature rise and
corresponding buckling modes.
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 6 51
a b
Fig. 4. Nondimensional maximal deflections versus pulse amplitude (a) and DLF (b) for different
volume fraction index n and pulse duration t tp � 0 (clamped plate).
a b
Fig. 5. Nondimensional maximal deflections versus pulse amplitude (a) and DLF (b) for different
volume fraction index n and pulse duration t tp � 0 5 0. (clamped plate).
a b
Fig. 6. Nondimensional maximal deflections versus pulse amplitude (a) and DLF (b) for different
volume fraction index n and pulse duration t tp � 0 (simply supported plate).
Comparing the curves presented in Figs. 4–7 and the results shown in Tables 4
and 5 it can be noticed that:
(i) the relations w h f DLF� ( ) for assumed boundary conditions and
temperature pulse duration are almost independent on volume fraction index n
(Figs. 4b, 5b, 6b, and 7b) but values of DLFcr determined on the basis of
Budiansky–Hutchinson criterion differs from 10 to 15%;
52 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 6
L. Czechowski and K. Kowal-Michalska
T a b l e 4
Static and Dynamic Critical Values of Temperature Increment for t tp � 0
Volume fraction
index n
Clamped plate Simply supported plate
�Tcr
st , K �Tcr
dyn , K DLFcr �Tcr
st , K �Tcr
dyn , K DLFcr
0 25.18 36.0 1.43 9.42 12.0 1.38
0.3 28.19 37.5 1.33 10.70 13.0 1.21
1.0 30.55 40.0 1.31 11.72 13.5 1.15
5.0 34.19 44.0 1.29 12.48 14.0 1.12
� 39.59 56.0 1.41 14.80 17.0 1.15
T a b l e 5
Static and Dynamic Critical Values of Temperature Increment for t tp � 0 5 0.
Volume fraction
index n
Clamped plate Simply supported plate
�Tcr
st , K �Tcr
dyn , K DLFcr �Tcr
st , K �Tcr
dyn , K DLFcr
0 25.18 67 2.66 9.42 32.5 3.45
0.3 28.19 70 2.48 10.70 – –
1.0 30.55 83 2.71 11.72 34.0 2.90
5.0 34.19 95 2.82 12.48 38.0 3.04
� 39.59 112 2.69 14.80 52.0 3.61
a b
Fig. 7. Nondimensional maximal deflections versus pulse amplitude (a) and DLF (b) for different
volume fraction index n and pulse duration t tp � 0 5 0. (simply supported plate).
(ii) for shorter pulse duration of temperature rise the character of curves
w h f DLF� ( ) changes and the values of DLFcr grow up (this effect is greater in
case of simply supported plates).
In Fig. 8 the comparison of dynamic responses of functionally graded plates
with volume fraction index n�1 (it is a specific case when material properties are
distributed linearly across the plate thickness – see Fig. 1) is shown. The influence
of boundary conditions and pulse duration of temperature rise on the character of
curves w h f DLF� ( ) can be easily seen. It is interesting to point out that for
clamped plate for t tp �0 5 0. the deflections are much smaller than for t tp � 0
(for the same value of DLF) and the curves intersect at the value DLF �6, while
for simply supported plates the curves for t tp �0 5 0. and t tp � 0 stay closer and
they intersect at the value of DLF close to 4.
Conclusions. In this paper, the analysis of dynamic behavior of FG plates
subjected to uniform temperature rise, applied dynamically as a pulse of finite
duration, have been conducted. This analysis has shown that similarly as for
mechanical pulse loadings the character of plate dynamic responses and dynamic
critical values of temperature increment strongly depend on pulse duration and – as
it is expected – on the boundary conditions. It should be marked that the
considerations have been conducted under the assumption that the material
properties are temperature independent and for small initial imperfection amplitude,
therefore the effect of imperfection sense (towards ceramic surface or metallic
surface) could be neglected. For larger values of initial imperfection amplitudes the
sense of imperfection can influence the critical values of pulse temperature
increment.
This study is supported by the Ministry of Science and Higher Education in
Poland – National Science Center Grant No. 2011/01/B/ST8/07441.
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 6 53
Static and Dynamic Buckling of Rectangular Functionally Graded Plates ...
Fig. 8. Comparison of dynamic nondimensional responses for two pulse duration and two boundary
conditions – volume fraction index n �1.
Ð å ç þ ì å
Ïðåäñòàâëåíî ÿâèùå âòðàòè ñò³éêîñò³ ï³ä ÷àñ ñòàòè÷íîãî ³ äèíàì³÷íîãî íàâàí-
òàæåííÿ (³ìïóëüñ ñê³í÷åííî¿ òðèâàëîñò³) ôóíêö³îíàëüíî-ãðà䳺íòíèõ ïëàñòèí
çà ð³âíîì³ðíîãî ï³äâèùåííÿ òåìïåðàòóðè. Ðîçãëÿíóòî øàðí³ðíî îïåðò³ àáî
çàêð³ïëåí³ (âçäîâæ êðîìîê) òîíê³ ïðÿìîêóòí³ ïëàñòèíè ç íåðóõîìèìè êðîì-
êàìè. Ïðèïóñêàºòüñÿ, ùî âëàñòèâîñò³ ìàòåð³àëó, ÿê³ ïëàâíî çì³íþþòüñÿ ïî
òîâùèí³, íå çàëåæàòü â³ä òåìïåðàòóðè. Ïðîâåäåíî äîñë³äæåííÿ äëÿ ð³çíèõ
çíà÷åíü êîåô³ö³ºíòà â³äíîñíîãî îá’ºìó ³ ð³âíîì³ðíîãî ïðèðîñòó òåìïåðàòóðè,
ïðåäñòàâëåíîãî ó âèãëÿä³ ïðÿìîêóòíîãî ³ìïóëüñó ñê³í÷åííî¿ òðèâàëîñò³.
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54 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 6
L. Czechowski and K. Kowal-Michalska
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Received 11. 04. 2013
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2013, ¹ 6 55
Static and Dynamic Buckling of Rectangular Functionally Graded Plates ...
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